Eco-evolutionary buffering: rapid evolution facilitates ...€¦ · 6/3/2016 · 26 2012; Fukami and Nakajima 2013). Founded in large part on MacArthur and Wilson’s (1967) theory

Eco-evolutionary buffering: rapid evolution facilitates

regional coexistence of species engaged in local priority

effects

Meike J. Wittmann∗, Tadashi Fukami†

June 3, 2016

Abstract

Priority effects, in which early-arriving species exclude competing species from local com-2

munities, are thought to enhance regional species diversity by promoting community diver-

gence. Theory suggests, however, that these same priority effects make it impossible for4

species to coexist in the region unless individuals are continuously supplied from an external

species pool, often an unrealistic assumption. Here we develop an eco-evolutionary hypoth-6

esis to solve this conundrum. Using a metacommunity model in which priority effects occur

via interspecific interference, we show that species can coexist regionally even in the absence8

of an external species pool if resistance to other species evolves rapidly under a trade-off

with competitive ability. With this trade-off, species become less resistant as they become10

regionally more common. Rare species can then invade and dominate some local patches

and consequently recover in regional frequency. Intriguingly, rare species recovery is possible12

even while local priority effects remain strong. This eco-evolutionary buffering of rare species

appears especially effective when many small communities are connected by infrequent dis-14

persal.

Keywords: priority effect, metacommunity, eco-evolutionary dynamics, competition, coexistence,16

species pool.

∗Department of Biology, Stanford University. Present address: Faculty of Mathematics, University of Vienna,

[email protected]†Department of Biology, Stanford University, [email protected]

1

certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. The copyright holder for this preprint (which was notthis version posted June 3, 2016. ; https://doi.org/10.1101/040659doi: bioRxiv preprint

https://doi.org/10.1101/040659

Introduction18

There is now ample evidence that the effects that species exert on one another in a local habitat

patch often depend on the order and initial abundance in which species arrive (Sutherland 1974;20

Drake 1991; Chase 2003). Known as priority effects (Slatkin 1974), such historical contingency

in local community assembly is increasingly recognized as a major factor influencing species22

diversity (Fukami 2015). In particular, recent research has shown that local priority effects can

enhance beta diversity, i.e., the variation in species composition among local communities, by24

driving communities onto divergent successional trajectories (e.g., Chase 2010; Martin and Wilsey

2012; Fukami and Nakajima 2013).26

Founded in large part on MacArthur and Wilson’s (1967) theory of island biogeography,

much of this research has traditionally assumed an external species pool (e.g., Post and Pimm28

1983; Lockwood et al. 1997; Morton and Law 1997). That is, immigrants entering local patches

are drawn from a regional pool whose species composition is static and is not influenced by local30

community dynamics. However, at large spatial and temporal scales, the regional pool consists of

immigrants originating from a collection of local patches themselves (Mittelbach and Schemske32

2015). In other words, the regional pool is not external, but internal (sensu Fukami 2005, 2015),

as depicted by the metacommunity concept (Leibold et al. 2004).34

To explain species diversity at these scales, it is therefore necessary to understand how a di-

verse species pool can be maintained as a collective result of local community dynamics. This36

task is particularly challenging when species engage in inhibitory priority effects, where species

that are initially common hinder colonization by competing species (Shurin et al. 2004). In many38

cases, species are likely to arrive at a newly created or disturbed patch in proportion to their re-

gional abundances within the metacommunity. This correspondence between regional frequency40

and arrival probability can result in neutral population dynamics at the regional scale (Taneyhill

2000), with all but one species eventually drifting to regional extinction. In other cases, regionally42

common species can benefit disproportionately from the local priority effects and regionally rare

species tend to become even rarer and eventually disappear from the region (Shurin et al. 2004).44

Thus, to maintain both local priority effects and a diverse regional pool of species, there has to

be a mechanism that buffers species from regional extinction. In a pioneering study, Shurin et al.46

(2004) suggested that spatial environmental heterogeneity could be one such mechanism. In their

specific model, patches differ in the concentrations of two essential resources. Two species may48

then engage in priority effects in patches with relatively balanced resource composition while

there is strict competitive exclusion of one or the other species in patches with more extreme50

resource composition. The extreme patches then serve as refuges from which species continue

2


https://doi.org/10.1101/040659

to disperse into patches where priority effects occur. In this sense, invoking spatial refuges is52

conceptually no different from assuming an external species pool. As a consequence, it remains

unknown whether regional species diversity can be maintained in the presence of local priority54

effects without recourse to an external species pool.

The goal of this paper is to suggest that this maintenance is possible when species traits56

responsible for priority effects evolve rapidly. Many previous studies of priority effects have

assumed that species traits are fixed (but see, e.g., Urban and De Meester 2009; Knope et al. 2012).58

Growing evidence indicates, however, that traits often evolve at a speed comparable to that of

ecological population dynamics (Thompson 1998; Schoener 2011). Effects of such rapid evolution60

(sensu Hairston et al. 2005) on species coexistence are emerging as an active area of research

(Lankau 2011). For example, Lankau (2009) and Vasseur et al. (2011) suggest that evolution along62

a trade-off between intra- and inter-specific competitive ability can prevent local extinction even

when there is some positive frequency dependence, i.e. a common-species advantage. However,64

these studies did not consider coexistence at the metacommunity level.

Here we use a simple metacommunity model inspired by empirical examples of priority66

effects to determine whether rapid evolution can allow coexistence in metacommunities when

species engage in local priority effects. Previous work that considered other types of spatial68

habitat structure (Vellend and Litrico 2008; Lankau 2009) calls into question whether rapid evo-

lution can promote coexistence when there are multiple local communities that only occasionally70

exchange immigrants rather than one well-mixed community. Moreover, even if rapid evolu-

tion does promote coexistence in a metacommunity, it might also promote coexistence in local72

communities and thereby eliminate local priority effects altogether. Thus, it is far from obvi-

ous whether and how local priority effects and regional species coexistence can be maintained74

simultaneously. Our model is designed to generate hypotheses on this fundamental question.

Methods76

Empirical motivation

In this study, we focus on inhibitory priority effects via interspecific interference, of which there78

are many empirical examples in microbes, animals, and plants. Microbes inhabiting floral nec-

tar, for example, appear to change the chemical properties of nectar in a way that makes it80

harder for other, late-arriving species to colonize (Peay et al. 2012; Vannette et al. 2013). This

type of self-serving habitat modification causes inhibitory priority effects. Similarly, in marine82

soft-bottom sediments, ghost shrimps and bivalves each modify grain size and oxygen content,

and each group thrives better in its self-modified environment (Peterson 1984; Knowlton 2004),84

3


https://doi.org/10.1101/040659

another case of inhibitory priority effects via habitat modification. In plant communities, local

positive feedbacks have been found to operate in some landscapes with interspersed patches of86

forest and heathland, mediated in this case by fire frequency and nutrient cycling (Petraitis and

Latham 1999; Odion et al. 2010). More generally, many species of microbes and plants engage in88

“chemical warfare” with their competitors, causing inhibitory priority effects by habitat modifi-

cation. Bacteria, for example, produce bacteriocins, compounds that inhibit or kill closely related90

strains or species, but do not affect the producing strain itself (Riley 1998). Many plants, in-

cluding invasive species can produce allelopathic chemicals that harm heterospecific individuals92

more than conspecifics (Bais et al. 2003; Callaway and Ridenour 2004).

Furthermore, empirical evidence suggests that traits involved in inhibitory priority effects94

often evolve rapidly along a trade-off with other aspects of fitness. For example, microbial resis-

tance to bacteriocins or antibiotics evolves so rapidly that it is a pressing medical and economic96

problem (Palumbi 2001; Baquero et al. 2011). But rapid evolution of microbial resistance often

comes at a cost such as reduced growth rate (Riley 1998), reduced competitive ability (Gagneux98

et al. 2006), or “collateral sensitivity” to other types of antimicrobials (Pal et al. 2015), at least

until compensatory mutations alleviate it (Riley 1998). Similarly, in some plants, such as species100

of Brassica, both allelotoxin production and growth rate can evolve rapidly, but along a trade-off

between the two traits (Lankau 2008; Lankau et al. 2009; Lankau 2011). Inspired by these em-102

pirical examples, we built a simple metacommunity simulation model with priority effects via

interspecific interference and evolution along trade-offs as key ingredients.104

Model

Overview106

Our simulation model is stochastic, individual-based, and considers a large number of patches.

The key processes are birth and death events, migration between patches, evolution of two traits108

involved in priority effects (general sensitivity to competition and sensitivity to interference by

other species), disturbance, and recolonization of patches (fig. 1, see also Table A1). In this model,110

local and regional dynamics are linked via migration and recolonization, making the species pool

internal rather than external. If regional coexistence is not possible for two species, it is unlikely112

to be possible with more species. We therefore focus on metacommunities consisting of two

species as a simplest case. However, to investigate the potential generality of our findings, we114

also explore some cases where up to 10 species are initially present in the metacommunity. All

species have the same dispersal ability. Our goal is to explore whether ecologically similar species116

engaging in local priority effects can coexist due to rapid evolution alone, that is in the absence of

4


https://doi.org/10.1101/040659

other coexistence mechanisms. We therefore assume that parameters are identical across patches.118

This means that there is no spatial environmental heterogeneity relevant to the coexistence of the

species, although patches may differ in all other respects.120

Death, birth, and migration

In our model, there are n patches, each of which is always fully occupied by a local community122

of k individuals, i.e., there is never any empty space in the patches. We assume patches are small

enough for inhabitants to be affected by interference effects from all other inhabitants. Time124

is continuous and death events occur at rate k in each patch such that one time unit roughly

corresponds to one generation. For each death event, we first pick an individual to be killed.126

Individuals are picked in proportion to their individual level of competitive pressure, C, which

results from actual competition and interference effects. As explained below, C may vary between128

individuals and its parameters evolve over time. Second, we choose an individual to reproduce

asexually. With probability 1− m, the parent individual comes from the same patch where the130

death event has just occurred (no migration), and with probability m it comes from a randomly

chosen other patch (migration). Finally, we randomly draw one of the k individuals in the132

selected patch as a parent, including, in the case of no migration, the individual that has just

died.134

Individual traits, trade-offs, and competitive pressure

Individuals are haploid and each of them has its own set of quantitative traits: general sensitivity136

to competition, G, and sensitivity to interference or habitat modification by each of the other

species j, Hj, with all trait values non-negative. A low value of G indicates a high general138

competitive ability (i.e., low mortality in a patch fully occupied by conspecifics), whereas a low

value of Hj indicates a high tolerance or resistance to species j. We assume that each species140

interferes or modifies the habitat in a unique way, for example by producing different toxins. The

strength of interference, e.g. the rate of toxin production, is assumed constant across individuals,142

species, and time.

The total competitive pressure, C, experienced by a focal individual is the sum of general

competitive effects of all individuals in the local community and the additional effects of inter-

ference. For a specific individual of species i having traits G and Hj

C = G · k︸︷︷︸general competition

+∑j 6=i

Hj · Nj︸︷︷︸interference from species j

, (1)

5


https://doi.org/10.1101/040659

where Nj is the local population size of species j. To avoid excessive indexing, we did not include144

indices for individual and deme number in (1). But keep in mind that individuals within the

same species may differ in G, Hj and consequently also C. Since C is smallest if all heterospecific146

abundances, Nj, are zero and increases with each Nj, interspecific competition is always stronger

than intraspecific competition and coexistence would be impossible in a single homogeneous148

community with fixed species traits. Only in the extreme case where all Hj are zero, interspecific

competition and interspecific competition would be equally strong. Competitive pressure only150

depends on the current community composition and not on past inhabitants of the patch. This

is realistic for all types of direct interspecific interference and also for many types of habitat152

modification, for example for toxins that rapidly decay or diffuse away.

In our full model, we assume a trade-off between each of the interspecific sensitivities, Hj,

and the general sensitivity to competition, G. For a specific member of species i:

G = gmin + ∑j 6=i

(1−

Hj

hmax

)y

· ∆g︸︷︷︸Gj

, (2)

where Gj is the cost in terms of general sensitivity to competition that the individual pays for154

reduced sensitivity to species j. The exponent y > 0 defines the shape of the trade-off curve. In

the main text, we focus on linear trade-offs (y = 1, fig. 1 b, but see Fig A6 for an exploration of156

nonlinear trade-offs).

To better understand the trade-off (fig. 1 b), let us consider the two extreme types for a given158

species (say species 1) sharing the landscape with one other species (species 2). On the one ex-

treme, we have individuals of species 1 that are fully susceptible to heterospecific interference160

(H2 = hmax) but have the smallest possible sensitivity to general competition, gmin. Those might

be individuals that do not invest in costly resistance pathways and are thus better at competing162

for resources. Such individuals experience little competitive pressure when surrounded by con-

specifics (gmink, see (1)) and thus survive longer and produce on average more offspring, but they164

experience high competitive pressure when surrounded by heterospecifics (gmink + hmax(k− 1)).

On the other extreme, we have species-1 individuals that are fully resistant to heterospecific in-166

terference (H2 = 0) but at the cost of the highest possible general sensitivity to competition,

gmin + ∆g. As an illustration, they might invest a lot of energy into defenses, which makes168

them weak competitors for resources. The total competition experienced by such individuals

does not depend on whether they are surrounded by conspecifics or heterospecifics, it is always170

(gmin + ∆g)k. If hmax > ∆g, which is fulfilled throughout this study, this is higher than what

individuals at the other extreme experience in conspecific environments, but lower than what172

they experience in heterospecific environments. In other words, there is a trade-off between

6


https://doi.org/10.1101/040659

intraspecific and interspecific competitive ability.174

Via mutations in individuals’ traits, populations can evolve along the trade-off (fig. 1 b).

Specifically, every new offspring experiences mutation, which adds a small, normally distributed176

increment with mean 0 and standard deviation d to the parent’s value of Gj. If there are multiple

interacting species, this is done independently for each of them. The values of Hj then follow178

from the trade-off equation. The parameter d can be interpreted as the sum over all relevant

loci of effect size and mutation rate. If a trait overshoots the minimum or maximum possible180

value, it is set to the minimum or maximum (0 or ∆g for the Gj, and 0 or hmax for the Hj).

Note that even in cases where one extreme of the trade-off is optimal, mutations will prevent182

the population from becoming monomorphic and the average trait values will reflect a balance

between selection pressure and mutations.184

To investigate the metacommunity consequences of trait evolution and the trade-off between

general sensitivity to competition and sensitivity to heterospecific interference, we compare the186

“trade-off evolution” scenario just described to three alternative scenarios. First, in the “neutral”

scenario (no trade-off, no evolution), all individuals in the metacommunity have the same general188

sensitivity to competition (G = 1) and are completely resistant to heterospecific interference

(Hj = 0 for all j). Second, in the “fixed trait” scenario (trade-off, but no evolution), there is no190

mutation such that all Gj and Hj stay at their initial values, by default at the midpoint of the

trade-off curve. Third, in the “independent evolution” scenario (evolution, but no trade-off), the192

Gj and Hj evolve independently. Mutations for Hj work exactly as described above for mutations

in Gj, but the standard deviation of mutational steps is multiplied by hmax/∆g to account for the194

different scale of the trait space. In the independent-evolution scenario, selection acts to reduce

both the Hj and the Gj.196

Patch disturbance and recolonization

As in Shurin et al. (2004), we assume that entire patches can be destroyed and recolonized. With-198

out such disturbance or when disturbance occurs at a smaller scale than local positive feedbacks,

the landscape may settle into a configuration where each patch is dominated by one species. The200

regional dynamics then come to a halt and species can coexist for extended periods of time, as

demonstrated by Molofsky et al. (1999, 2001) and Molofsky and Bever (2002) for spatially ex-202

plicit models. However, since there is no disturbance to initiate new rounds of local community

assembly, priority effects will no longer be operating.204

In our model, disturbance events occur at rate 1/τ in each patch, i.e. τ is the average time

between disturbance events (fig. 1 c). When a patch is disturbed, all its inhabitants die. The206

patch is then recolonized immediately by a founder population of size f , consisting of offspring

7


https://doi.org/10.1101/040659

of individuals elsewhere in the metacommunity. For the interplay between local and regional208

dynamics that is central to this study, it may be important where in the metacommunity the f

recolonizing individuals originate. We therefore use two recolonization variants, the “propagule-210

pool model” and the “migrant-pool model” (Slatkin 1977). In the propagule-pool model, all f

recolonizing individuals come from the same randomly drawn other patch, such that they reflect212

the local species composition of one other patch. Under this model, a regionally rare species

will often be absent among the initial colonizers of a patch but can sometimes be common.214

In the migrant-pool model, by contrast, the source patch is drawn independently for each of

the f individuals such that the recolonizing individuals are more representative of the regional216

species composition. In the presence of local priority effects, migrant-pool recolonization should

therefore benefit the regionally more common species.218

After drawing the source patch(es), parents in the source patch(es) are drawn independently

and with replacement to produce an offspring for the founder population. The founder popula-220

tion of size f then gives rise to a population of size k in one round of reproduction, i.e. for each of

the k places to be filled, a parent individual is drawn independently and with replacement from222

the f recolonizers. Mutations can happen in each reproduction event during recolonization.

Simulations224

To explore the parameter space for two-species metacommunities, we generated 150 combina-

tions of seven parameters (number of patches n, local community size k, migration probability226

m, maximum sensitivity to heterospecific interference hmax, the standard deviation of mutation

effect sizes d, average time between disturbance events τ, and founder population size f , see228

fig. 1) by drawing independently from the distributions given in Table A1. In addition, recolo-

nization followed either the propagule-pool or migrant-pool model, with equal probability. The230

remaining parameters were fixed to the values given in Table A1.

In choosing these parameter ranges, we kept in mind both computational efficiency and bio-232

logical relevance. Because individual-based simulations are computationally intensive, we were

limited to simulating metacommunities with on the order of 10,000 individuals. Given this con-234

straint, we decided to focus on metacommunities with hundreds of small patches rather than

metacommunities with few larger patches. With our parameter choice, we expect fast and highly236

stochastic dynamics at the local scale, compared to slower and more deterministic dynamics at

the regional scale. Also, we chose relatively frequent migration to ensure that evolution should238

occur at the regional scale rather than at the local scale (Lenormand 2002). We want to avoid evo-

lution at the local scale because it might lead to local coexistence and thereby eliminate priority240

effects. In summary, our metacommunity set-up should be conducive to achieving both regional

8


https://doi.org/10.1101/040659

coexistence and local priority effects, if this is possible at all. Metacommunities of herbaceous242

plants or sessile aquatic animals might be the best empirical examples for metacommunities in

this parameter range (see also Mouquet and Loreau 2002; Fukami and Nakajima 2013; Zee and244

Fukami 2015). However, as we discuss below, some of our results may also apply to other types

of metacommunities and even microorganisms.246

For each parameter combination, we ran 10 replicate simulations. At the beginning of each

simulation run, we assigned individuals randomly to species. All individuals in the metacommu-248

nity initially had the same general sensitivity to competition Gj = g0 for all j and corresponding

values of Hj. Storing regional frequencies every ten time units, we then ran each simulation for250

10,000 time steps (generations).

Code and scripts252

We provide C++ code for the simulation program in Online Appendix B, and R scripts to initiate

and analyze the simulation studies in Online Appendix C.254

Results

Probability of coexistence and outcome types256

For a first impression of whether regional coexistence of two species is stable, we considered

the proportion of replicate metacommunities that still contained both species at the end of a258

simulation run. Across all of the simulation runs we carried out, regional species coexistence

was most likely under trade-off evolution. Specifically, averaged over all 150 randomly generated260

parameter combinations, the two species coexisted until the end of the simulation in 50 % of the

cases under trade-off evolution, whereas they coexisted in only 25, 9, and 18 % of the cases in the262

neutral, fixed-trait, and independent-evolution scenarios, respectively (fig. 2).

In some cases where species coexisted until the end of the simulation, the regional dynamics264

may be slow and there may simply not have been enough time for one species to go extinct.

For the regional species pool to be actually stable in the long run, a regionally rare species must266

on average become more common again. To determine whether or not this is the case in our

simulations, we computed average changes in regional frequency for various frequency ranges.268

Based on these results, we then assigned a parameter combination to one of four frequency-

dependence classes (see fig. 2, Online Appendix A.1). Under positive frequency-dependence,270

regionally more common species have an advantage and regionally rare species are expected to

become even more rare and eventually go extinct. Such parameter combinations do not allow272

9


https://doi.org/10.1101/040659

for regional coexistence. Under negative-frequency dependence, by contrast, regionally rare

species have an advantage and are expected to become more common again. Coexistence at274

intermediate frequencies is expected. Under complex frequency-dependence, very rare species

have an advantage and are expected to increase in regional frequency, but moderately rare species276

are at a disadvantage. Therefore, we do expect coexistence, but in an asymmetric fashion where

one species is relatively rare in the landscape and the other common. Finally, we assigned278

parameter combinations without clear frequency dependence to the quasi-neutrality class. In

this class, we do not expect long-term coexistence because one or the other species will drift to280

extinction. Because we did not want to erroneously conclude that regional coexistence is possible

when it is not, we were very conservative in assigning scenarios to one of the “interesting” classes282

and assigned many scenarios to the quasi-neutral class even if they showed a slight tendency for

rare species to become more common again and diversity to be maintained for longer than under284

strict neutrality.

Under fixed traits or independent evolution, some parameter combinations were classified286

as quasi-neutral, but most fell into the positive frequency dependence class (fig. 2). That is, the

regionally rare species tended to become even rarer and eventually dropped out of the metacom-288

munity. The results on frequency-dependence and coexistence were qualitatively robust with

respect to the point along the trade-off to which the traits were fixed (fig. A7). With increasing290

general sensitivity to competition, coexistence and quasi-neutrality became only slightly more

common. Under trade-off evolution, many parameter combinations still exhibited positive fre-292

quency dependence or quasi-neutrality, but a sizable number of parameter combinations (19 out

of 150) were classified as having either complex or negative frequency dependence, meaning that294

there was a clear tendency for the regionally rare species to become more common again and that

such systems are stable. These coexistence outcomes did not appear in any of the other scenarios,296

except for two independent-evolution scenarios that showed negative frequency dependence.

As expected, coexistence until the end of the simulation was most likely under complex or298

negative frequency dependence (100 % in both cases) and least likely under positive frequency

dependence (fig. 2). Parameter combinations that were classified as quasi-neutral under trade-off300

evolution had a substantially larger coexistence probability than quasi-neutral parameter combi-

nations in the fixed-trait or independent-evolution. This indicates that our conservative classi-302

fication scheme may indeed have classified many scenarios as quasi-neutral that do have some

tendency to maintain diversity.304

For each of the four outcome classes that occurred under trade-off evolution, fig. 3 shows

time series for one representative parameter combination, together with time series for the cor-306

responding three other scenarios for comparison. As expected, coexistence under negative fre-

10


https://doi.org/10.1101/040659

quency dependence was symmetric, with both species at regional frequencies around 0.5 (fig. 3 l).308

Additional simulations with initial regional frequencies closer to 0 or 1 suggest that this symmet-

ric coexistence equilibrium is indeed globally attracting. By contrast, under complex frequency310

dependence, coexistence was asymmetric, with one species more abundant in the region than the

other (fig. 3 p). Within each outcome class, parameter combinations differed in the magnitude of312

fluctuations around the respective mean frequencies.

Evolutionary dynamics314

In the negative frequency dependence class under trade-off evolution (fig. 4 e), both species re-

mained highly tolerant of interspecific interference (low Hj) and therefore had a high sensitivity316

to general competition (high G) compared to species under the quasi-neutral and positive fre-

quency dependent classes (fig. 4 a–d). However, common species tended to be more sensitive to318

interspecific interference (fig. 4 f). In the corresponding independent evolution scenario, both G

and Hj were low at equilibrium but still responded weakly to regional frequency (fig. A3). Under320

complex frequency dependence (fig. 4 g,h), regionally rare species remained relatively resistant

to the other species (low Hj), whereas common species evolved to become more competitive322

(lower G) and thus more sensitive to the other species (higher Hj). When species frequencies

fluctuated, trait values fluctuated also, particularly in the regionally common species (fig. 4 g,h).324

Strength of priority effects

To quantify priority effects, we explored how often the most abundant species among the initial326

colonists of a newly disturbed patch retained its dominant status over time. Specifically, we

took all intervals between local disturbance events for all the patches in a simulation run and328

computed the proportion of such intervals for which the same species was locally most common

both at the beginning and at the end of the interval. This proportion tends to be higher if the330

regional pool has an uneven composition because different immigrants to a focal patch then

belong to the same species more often. To make the results comparable across scenarios, we332

therefore controlled for regional frequency. For this, we used regional frequency bins of size 0.025

and computed the strength of priority effects separately for each bin. We performed the same334

analyses for the corresponding neutral scenario. If the resulting value for a certain frequency

range is larger than in neutral scenario, we take this as evidence for inhibitory priority effects.336

We found that local priority effects under trade-off evolution were generally stronger than in

the corresponding neutral simulations, at least while both species had a regional frequency of338

20 % or more (gray regions in fig. 3, see also fig. A4). However, when one species dominated

11


https://doi.org/10.1101/040659

the metacommunity, priority effects were often nearly identical or slightly weaker in strength340

compared to those in the neutral scenario (white regions in fig. 3, left side of fig. A4 d). For a

given regional frequency, there were no substantial differences in the strength of priority effects342

between trade-off evolution, independent evolution, and fixed-trait scenarios (fig. A4 a–d). In the

long run, however, priority effects were visible only in scenarios with evolution and negative or344

complex frequency dependence (e.g., fig. 3 l,p). In the remaining cases, priority effects eventually

disappeared along with species diversity (e.g., fig. 3 g,h).346

Sensitivity to individual parameters

To better understand how the joint maintenance of regional diversity and local priority effects348

arises and how it depends on the model parameters, we focused on one scenario with negative-

frequency dependence (fig. 3 l) and ran additional simulations varying two parameters system-350

atically at a time while keeping the other parameters constant. Here we ran 50 replicates per

parameter combination.352

We found that negative frequency dependence generally occurs for small local community

sizes k, with quasi-neutrality or complex frequency dependence at intermediate values of k, and354

positive frequency dependence for large k (fig. 5). The range of k values with negative frequency

dependence was largest with small migration rates m (fig. 5 a), large standard deviations of356

mutational effects d (fig. 5 b), and large maximum sensitivity to interference hmax (fig. 5 d). The

average time between disturbance events, τ, did not have a strong effect on the distribution of358

outcome types (fig. 5 c). The region of parameter space with stable coexistence grew slightly

with increasing convexity of the trade-off (fig. A6).360

For additional information on the role of the various parameters, we also considered their

distributions among the outcome classes in the random parameter study. For k and m, the re-362

sults are consistent with those described in the previous paragraph, but not for d and τ (fig. A5),

indicating that these parameters may have opposing effects for different combinations of the364

other parameters. We also found that the complex and positive frequency dependence classes

contained more parameter combinations with migrant-pool recolonization, whereas the negative366

and quasi-neutral classes contained more parameter combinations with propagule-pool recolo-

nization (fig. A5 h). This finding is consistent with our expectation that migrant-pool recoloniza-368

tion is beneficial for the regionally common species.

12


https://doi.org/10.1101/040659

More than two species370

For two parameter combinations with clear regional coexistence, we additionally ran simulations

with 3, 4, 5, and 10 species, this time for 20,000 generations. Under a parameter combination with372

negative frequency dependence (third row in fig. 3), three species coexisted until the end of the

simulation in all cases under trade-off evolution, and most of the replicate metacommunities that374

started with four, five, or ten species maintained four species until the end (fig. 6). In all cases,

regional species diversity in the neutral and fixed-trait scenarios decayed more rapidly than in376

the trade-off evolution scenario. But as in the two-species case, some independent-evolution

simulations also resulted in coexistence. Again, priority effects were usually stronger under378

trade-off evolution than under neutrality when controlling for regional community composition

as measured by Simpson’s diversity index (fig. A9). The slightly more complex outcomes in a380

parameter combination with complex frequency-dependence (last row in fig. 3) are explored in

Online Appendix A.4.382

Discussion

Taken together, our results suggest a new hypothesis for the maintenance of a multi-species re-384

gional pool in the presence of local priority effects. This “eco-evolutionary buffering” hypothesis

can be summarized as follows. Assuming resistance to heterospecific interference is costly and386

evolves along a trade-off with general competitive ability, the strength and direction of selec-

tion will depend on regional frequencies of species. When other species are common in the388

region, members of a species experience strong selective pressure to be tolerant of heterospecific

interference. However, as a species approaches regional dominance as a cumulative result of390

local priority effects, its members lose the costly resistance to other species (e.g., cycles in fig. 4).

Exploiting this loss of tolerance, other species recover in abundance.392

Previous studies have explored similar coexistence mechanisms in a single local community,

rather than in a metacommunity (Levin 1971; Leon 1974; Pease 1984; Vasseur et al. 2011). Here we394

have shown that such coexistence is also possible in metacommunities consisting of many small

patches and in the face of demographic and genetic stochasticity, important factors especially in396

the recovery of rare species. Moreover, while previous studies have focused on two-species sys-

tems, our results show that eco-evolutionary buffering can stabilize a regional pool of at least up398

to four species (fig. 6). Most importantly, however, we have shown that even if metacommunities

evolve toward traits that allow for regional coexistence, local coexistence does not necessarily400

follow. Neither is one species competitively superior at any time. Rather strong local priority

effects are often maintained (figs. A4 and A9). Thus our new hypothesis can explain how multi-402

13


https://doi.org/10.1101/040659

ple species that engage in local priority effects can stably co-exist even under an internal species

pool and even in the absence of environmental heterogeneity across local habitat patches.404

Types and requirements of eco-evolutionary buffering

Our results indicate that eco-evolutionary buffering can result in either negative frequency de-406

pendence where the regionally rare species is always favored and coexistence is symmetric, or

complex regional frequency dependence where only very rare species have an advantage and co-408

existence is generally asymmetric (fig. 2). Negative frequency dependence is particularly likely

when local patches are small and when they are connected by infrequent migration (figs. 5 and410

A5). Under these conditions, a small evolutionary response seems sufficient to stabilize regional

coexistence (fig. 4 e,f). In Online Appendix A.5, we offer a detailed heuristic explanation for this412

observation. Briefly, under small community size and low migration rate, the variance in local

community size relative to its expectation is large. Hence many individuals of regionally rare414

species are in patches where they are locally common. Therefore, these individuals do not suffer

much from local interference by the regionally common species, and a small evolutionary change,416

e.g., a slight shift in the balance between mutations and selection, is apparently sufficient to tip

the balance in favor of the regionally rare species. This explanation is consistent with the obser-418

vation that, for small local communities and rare migration, even some independent-evolution

scenarios exhibit negative frequency dependence.420

In other parts of parameter space in our model, a slight shift in mutation-selection balance

does not appear to be sufficient to give an advantage to regionally rare species. We speculate422

that they have to become rare enough for the direction of selection to change in the common

species. In some of these cases, rapid evolution buffers rare species from extinction via complex424

frequency dependence. In other cases, either because evolution is too slow or because selection

changes direction only at an extreme regional frequency, positive frequency dependence or quasi-426

neutrality ensues and species cannot coexist in the long run. However, the proportion of species

that coexisted until the end of the simulation was still elevated in the trade-off evolution scenario428

compared to the other scenarios (fig. 2). We suspect that deterministic models with simplified

genetic assumptions might indicate stable coexistence for many of the cases that we classified430

as unstable in the face of demographic and genetic stochasticity and using our conservative

classification scheme.432

Under many of the parameter combinations leading to complex frequency dependence, re-

gional frequencies oscillate (see, e.g., fig. 3 p). Times during which the rare species has an434

advantage and priority effects are weak alternate with times during which priority effects are

strong and the common species has an advantage. Under negative frequency dependence, how-436

14


https://doi.org/10.1101/040659

ever, local priority effects can be strong while at the same time regionally rare species have an

advantage. We propose that a slight asymmetry in the strength of interspecific competition and438

local priority effects is key to understanding why priority effects remain strong. Each species

might benefit from priority effects when it is substantially more common locally, but in patches440

where multiple species have a similar local frequency, regionally rare species might have an ad-

vantage even if they are slightly less common locally. For example, if local communities consist442

of six individuals, each species may have an advantage when it has four or more individuals

in the patch, but if both species have three individuals then the regionally rare species has an444

advantage (see fig. A11 and Online Appendix A.5 for more detail).

Based on our explorations of the parameter space (figs. 5 and A5, see Online Appendix A.3446

for more detail), we expect to observe eco-evolutionary buffering at intermediate frequencies

(negative frequency dependence) in metacommunities consisting of numerous local patches with448

only few individuals per patch, which may be possible in plants or sessile aquatic organisms. The

requirements for eco-evolutionary buffering by complex frequency dependence are less restric-450

tive and may be fulfilled even in some microbial communities. Although spatial heterogeneity

is obviously common in natural systems, often a large number of species coexists in a land-452

scape with surprisingly homogeneous environmental conditions (e.g., Sogin et al. 2006), and

eco-evolutionary buffering may contribute to explaining such observations.454

Future directions

This study is only a proof of concept, a first step toward understanding the role of eco-evolutionary456

buffering in the maintenance of species diversity. Questions that should be addressed in future

work include (1) how the total number of species that can coexist in a landscape depends on458

ecological and evolutionary parameters, (2) whether eco-evolutionary buffering also works for

diploid sexual organisms and for parameters relevant to microbial systems, and (3) how eco-460

evolutionary buffering interacts with spatial and temporal environmental heterogeneity to affect

regional coexistence. In addition, it would be useful to consider evolution of the rate of habitat462

modification in addition to the sensitivity to habitat modification. A mutation that reduces sensi-

tivity to heterospecific habitat modification directly reduces the death rate of the focal individual.464

In contrast, a mutation that increases the rate of habitat modification first influences the death

rate of heterospecific individuals. Indirectly, the mutation may then benefit the focal individual,466

but also non-mutated conspecifics that do not pay the fitness cost of an increased rate of habitat

modification. Hence habitat modification can be an altruistic trait in this case, and cheating may468

play a role in regional species coexistence. It is unclear at this point how readily eco-evolutionary

buffering occurs in these circumstances.470

15


https://doi.org/10.1101/040659

Besides the specific eco-evolutionary buffering mechanism we have studied in this paper, a

number of other mechanisms could potentially buffer regional diversity in the presence of pri-472

ority effects. Simple patch-occupancy models seem to suggest that, by virtue of spatial structure

alone, two identical competitors can coexist in a region even if there is some local inhibition474

(Slatkin 1974; Hanski 1983). However, this requires doubly-occupied patches to send out the

same number of colonists of both species (Taneyhill 2000), an assumption that gives an “unfair”476

advantage to the regionally rare species and even leads to logical inconsistencies (Wang et al.

2005). We list three more promising examples for possible buffering mechanisms. First, a preda-478

tor that forages on a regional scale may either exhibit behavioral plasticity or evolve rapidly

to preferentially prey on regionally common species (e.g., Hughes and Croy 1993). Second, if480

patches differ in environmental conditions, regionally rare species may be better at evolutionary

monopolization of patches (Urban and De Meester 2009; De Meester et al. 2016) as they suffer482

less from the inflow of maladapted migrants (Lankau 2011). Finally, at a long evolutionary time

scale, any factor that accelerates speciation rate would help to maintain a speciose regional pool.484

Speciation rate itself may be affected by local priority effects (Fukami et al. 2007). Interactive

effects of speciation and priority effects on the generation and maintenance of species pools are486

a particularly interesting topic for future research.

Of course, empirical tests should accompany theoretical exploration of the mechanisms and488

conditions for eco-evolutionary buffering. Because of computational constraints, we were not

able to explore the parts of parameter space directly relevant to microbes in our simulation. How-490

ever, because many microbes have short generation times, they may be particularly amenable to

experimental tests of the eco-evolutionary buffering hypothesis. In such an experiment, spatial492

conditions (number of patches, migration rate, disturbance and recolonization parameters) could

be manipulated (e.g., Tucker and Fukami 2014; Pu and Jiang 2015). Other parameters, like the494

shape of the trade-off and the mutation parameters, could be measured in laboratory assays. One

would then need to monitor regional frequencies over time and regularly assess the traits of the496

various species in sensitivity assays.

Conclusion498

We have shown that species engaged in local priority effects can coexist in a metacommunity if

their sensitivity to heterospecific interference evolves rapidly, especially under a trade-off with500

competitive ability. Rapid evolution can lead to one globally attracting coexistence state, with

local priority effects maintained at all times, or alternative attractors where species coexist re-502

gionally, but fluctuate around different frequencies, with local priority effects occurring only at

16


https://doi.org/10.1101/040659

times when both species are relatively common. The eco-evolutionary buffering hypothesis we504

have developed here provides the first mechanism, to our knowledge, that can maintain regional

species diversity in the presence of local priority effects even under an internal species pool.506

Acknowledgments

For discussion and comments, we thank members of the community ecology group at Stanford,508

particularly Luke Frishkoff, Po-Ju Ke, Devin Leopold, Erin Mordecai, and Rachel Vannette, as

well as Angela Brandt, Joachim Hermisson, Kotaro Kagawa, Mike McLaren, Akira Mori, Pleuni510

Pennings, Dmitri Petrov, and anonymous reviewers. MJW acknowledges fellowships from the

Stanford Center for Computational Evolutionary and Human Genomics (CEHG) and from the512

Austrian Science Fund (FWF, M 1839-B29). This work was also supported by the NSF (DEB

1149600) and Stanford University’s Terman Fellowship. Simulations were performed on Stan-514

ford’s FarmShare Cluster and on the Vienna Scientific Cluster (VSC).

References516

Bais, H. P., R. Vepachedu, S. Gilroy, R. M. Callaway, and J. M. Vivanco. 2003. Allelopathy and

exotic plant invasion: From molecules and genes to species interactions. Science 301:1377–1380.518

Baquero, F., T. M. Coque, and F. de la Cruz. 2011. Ecology and evolution as targets: the need

for novel eco-evo drugs and strategies to fight antibiotic resistance. Antimicrobial Agents and520

Chemotherapy 55:3649–3660.

Callaway, R., and W. Ridenour. 2004. Novel weapons: invasive success and the evolution of522

increased competitive ability. Frontiers in Ecology and the Environment 2:436–443.

Chase, J. M. 2003. Community assembly: when should history matter? Oecologia 136:489–498.524

———. 2010. Stochastic community assembly causes higher biodiversity in more productive

environments. Science 328:1388–1391.526

De Meester, L., J. Vanoverbeke, L. J. Kilsdonk, and M. C. Urban. 2016. Evolving perspectives on

monopolization and priority effects. Trends in Ecology & Evolution 31:136–146.528

Drake, J. A. 1991. Community-assembly mechanics and the structure of an experimental species

ensemble. The American Naturalist 137:1–26.530

17


https://doi.org/10.1101/040659

Fukami, T. 2005. Integrating internal and external dispersal in metacommunity assembly: pre-

liminary theoretical analyses. Ecological Research 20:623–631.532

———. 2015. Historical contingency in community assembly: Integrating niches, species pools,

and priority effects. Annual Review of Ecology Evolution and Systematics 46:1–23.534

Fukami, T., H. J. E. Beaumont, X.-X. Zhang, and P. B. Rainey. 2007. Immigration history controls

diversification in experimental adaptive radiation. Nature 446:436–439.536

Fukami, T., and M. Nakajima. 2013. Complex plant-soil interactions enhance plant species diver-

sity by delaying community convergence. Journal of Ecology 101:316–324.538

Gagneux, S., C. D. Long, P. M. Small, T. Van, G. K. Schoolnik, and B. J. M. Bohannan. 2006. The

competitive cost of antibiotic resistance in Mycobacterium tuberculosis. Science 312:1944–1946.540

Hairston, N. G., S. P. Ellner, M. A. Geber, T. Yoshida, and J. A. Fox. 2005. Rapid evolution and

the convergence of ecological and evolutionary time. Ecology Letters 8:1114–1127.542

Hanski, I. 1983. Coexistence of competitors in patchy environment. Ecology 64:493–500.

Hughes, R., and M. Croy. 1993. An experimental analysis of frequency-dependent predation544

(switching) in the 15-spined stickleback, Spinachia spinachia. Journal of Animal Ecology 62:341–

352.546

Knope, M. L., S. E. Forde, and T. Fukami. 2012. Evolutionary history, immigration history, and

the extent of diversification in community assembly. Frontiers in Microbiology 2:273.548

Knowlton, N. 2004. Multiple ”stable” states and the conservation of marine ecosystems. Progress

in Oceanography 60:387–396.550

Lankau, R. 2008. A chemical trait creates a genetic trade-off between intra- and interspecific

competitive ability. Ecology 89:1181–1187.552

Lankau, R. A. 2009. Genetic variation promotes long-term coexistence of Brassica nigra and its

competitors. The American Naturalist 174:E40–E53.554

———. 2011. Rapid evolutionary change and the coexistence of species. Annual Review of

Ecology Evolution and Systematics 42:335–354.556

Lankau, R. A., V. Nuzzo, G. Spyreas, and A. S. Davis. 2009. Evolutionary limits ameliorate

the negative impact of an invasive plant. Proceedings of the National Academy of Sciences558

106:15362–15367.

18


https://doi.org/10.1101/040659

Leibold, M. A., M. Holyoak, N. Mouquet, P. Amarasekare, J. M. Chase, M. F. Hoopes, R. D.560

Holt, J. B. Shurin, R. Law, D. Tilman, M. Loreau, and A. Gonzalez. 2004. The metacommunity

concept: a framework for multi-scale community ecology. Ecology Letters 7:601–613.562

Lenormand, T. 2002. Gene flow and the limits to natural selection. Trends in Ecology & Evolution

17:183–189.564

Leon, J. A. 1974. Selection in contexts of interspecific competition. The American Naturalist

108:739–757.566

Levin, B. R. 1971. Operation of selection in situations of interspecific competition. Evolution

25:249–264.568

Lockwood, J. L., R. D. Powell, M. P. Nott, and S. L. Pimm. 1997. Assembling ecological commu-

nities in time and space. Oikos 80:549–553.570

MacArthur, R. H., and E. O. Wilson. 1967. The theory of island biogeography. Princeton Univer-

sity Press, Princeton, USA.572

Martin, L. M., and B. J. Wilsey. 2012. Assembly history alters alpha and beta diversity, exotic-

native proportions and functioning of restored prairie plant communities. Journal of Applied574

Ecology 49:1436–1445.

Mittelbach, G. G., and D. W. Schemske. 2015. Ecological and evolutionary perspectives on com-576

munity assembly. Trends in Ecology & Evolution 30:241–247.

Molofsky, J., and J. D. Bever. 2002. A novel theory to explain species diversity in landscapes:578

positive frequency dependence and habitat suitability. Proceedings of the Royal Society B-

Biological Sciences 269:2389–2393.580

Molofsky, J., J. D. Bever, and J. Antonovics. 2001. Coexistence under positive frequency depen-

dence. Proceedings of the Royal Society B-Biological Sciences 268:273–277.582

Molofsky, J., R. Durrett, J. Dushoff, D. Griffeath, and S. Levin. 1999. Local frequency dependence

and global coexistence. Theoretical Population Biology 55:270–282.584

Morton, R. D., and R. Law. 1997. Regional species pools and the assembly of local ecological

communities. Journal of Theoretical Biology 187:321–331.586

Mouquet, N., and M. Loreau. 2002. Coexistence in metacommunities: The regional similarity

hypothesis. The American Naturalist 159:420–426.588

19


https://doi.org/10.1101/040659

Odion, D. C., M. A. Moritz, and D. A. DellaSala. 2010. Alternative community states maintained

by fire in the Klamath Mountains, USA. Journal of Ecology 98:96–105.590

Pal, C., B. Papp, and V. Lazar. 2015. Collateral sensitivity of antibiotic-resistant microbes. Trends

in Microbiology 23:401–407.592

Palumbi, S. R. 2001. Humans as the world’s greatest evolutionary force. Science 293:1786–1790.

Pease, C. M. 1984. On the evolutionary reversal of competitive dominance. Evolution 38:1099–594

1115.

Peay, K. G., M. Belisle, and T. Fukami. 2012. Phylogenetic relatedness predicts priority effects in596

nectar yeast communities. Proceedings of the Royal Society B-Biological Sciences 279:749–758.

Peterson, C. H. 1984. Does a rigorous criterion for environmental identity preclude the existence598

of multiple stable points. The American Naturalist 124:127–133.

Petraitis, P. S., and R. E. Latham. 1999. The importance of scale in testing the origins of alternative600

community states. Ecology 80:429–442.

Post, W. M., and S. L. Pimm. 1983. Community assembly and food web stability. Mathematical602

Biosciences 64:169–192.

Pu, Z., and L. Jiang. 2015. Dispersal among local communities does not reduce historical contin-604

gencies during metacommunity assembly. Oikos 124:1327–1336.

Riley, M. A. 1998. Molecular mechanisms of bacteriocin evolution. Annual Review of Genetics606

32:255–278.

Schoener, T. W. 2011. The newest synthesis: Understanding the interplay of evolutionary and608

ecological dynamics. Science 331:426–429.

Shurin, J., P. Amarasekare, J. Chase, R. Holt, M. Hoopes, and M. Leibold. 2004. Alternative stable610

states and regional community structure. Journal of Theoretical Biology 227:359–368.

Slatkin, M. 1974. Competition and regional coexistence. Ecology 55:128–134.612

———. 1977. Gene flow and genetic drift in a species subject to frequent local extinctions. Theo-

retical Population Biology 12:253–262.614

Sogin, M. L., H. G. Morrison, J. A. Huber, D. M. Welch, S. M. Huse, P. R. Neal, J. M. Arrieta, and

G. J. Herndl. 2006. Microbial diversity in the deep sea and the underexplored “rare biosphere”.616

Proceedings of the National Academy of Sciences 103:12115–12120.

20


https://doi.org/10.1101/040659

Sutherland, J. P. 1974. Multiple stable points in natural communities. The American Naturalist618

108:859–873.

Taneyhill, D. E. 2000. Metapopulation dynamics of multiple species: The geometry of competition620

in a fragmented habitat. Ecological Monographs 70:495–516.

Thompson, J. N. 1998. Rapid evolution as an ecological process. Trends in Ecology & Evolution622

13:329–332.

Tucker, C. M., and T. Fukami. 2014. Environmental variability counteracts priority effects to624

facilitate species coexistence: evidence from nectar microbes. Proceedings of the Royal Society

B-Biological Sciences 281:20132637.626

Urban, M. C., and L. De Meester. 2009. Community monopolization: local adaptation enhances

priority effects in an evolving metacommunity. Proceedings of the Royal Society B-Biological628

Sciences 276:4129–4138.

Vannette, R. L., M.-P. L. Gauthier, and T. Fukami. 2013. Nectar bacteria, but not yeast,630

weaken a plant–pollinator mutualism. Proceedings of the Royal Society B-Biological Sciences

280:20122601.632

Vasseur, D. A., P. Amarasekare, V. H. W. Rudolf, and J. M. Levine. 2011. Eco-evolutionary dynam-

ics enable coexistence via neighbor-dependent selection. The American Naturalist 178:E96–634

E109.

Vellend, M., and I. Litrico. 2008. Sex and space destabilize intransitive competition within and636

between species. Proceedings of the Royal Society B-Biological Sciences 275:1857–1864.

Wang, Z.-L., D.-Y. Zhang, and G. Wang. 2005. Does spatial structure facilitate coexistence of638

identical competitors? Ecological Modelling 181:17–23.

Zee, P. C., and T. Fukami. 2015. Complex organism–environment feedbacks buffer species diver-640

sity against habitat fragmentation. Ecography 38:370–379.

21


https://doi.org/10.1101/040659

(a)

n patches

s species(default 2)

k individualsper patch

Migration probability m

m/(n-1)

m/(n-1)m/(n-1)

m/(n-1)

m/(n-1)

(c)Disturbance on averageevery τ time units

f founders arrive

(b)

Sensitivity to species j, Hj

Gen

eral

sen

sitiv

ity t

o co

mpe

titio

n, G

gmin

gmin

+ Δg

hmax

0

Distribution of mutationeffectsstandard deviation d

Good intraspecific competitor

Good interspecific competitor

Time

Figure 1: Illustration of the metacommunity model with its key parameters and processes. (a)

Landscape set-up. In the actual simulations, the number of patches is on the order of hundreds.

(b) Trade-off between individual competitive traits and mutation process. It is assumed here that

there are just two species such that G = gmin + Gj. (c) Patch disturbance and recolonization. See

Table A1 for sampling distributions for the various parameters in our simulation experiments.

22


https://doi.org/10.1101/040659

Fixe

d

Inde

pend

ent

Trad

e−of

fTotal

Negative

Complex

Quasi−neutral

Positive

Frequency−dependence class

150 150 150

0 2 5

0 0 14

18 29 52

132 119 79 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Scenarios

Proportion ofcoexistence

0

0.2

0.4

0.6

0.8

1

10Regional

frequency

Expect

ed c

han

ge in r

egio

nal fr

equency

0.5

+0-

+0-

+0-

+0-

Figure 2: Number of parameter combinations under fixed parameters, independent evolution,

and trade-off evolution that were assigned to the different regional frequency classes, and the

corresponding proportion of replicates in which both species coexisted until the end of the simu-

lation. The frequency-dependence cartoons depict the average change in the regional frequency

of a focal species in various frequency ranges between 0 and 1. Filled circles indicate what would

be stable equilibria in a deterministic system, and open circles denote unstable equilibria. Ar-

rows indicate the expected direction of change. See Online Appendix A.1 for more detail on the

classification of regional frequency dependence.

23


https://doi.org/10.1101/040659

(a) Quasi−neutral0.

00.

51.

0(b) Quasi−neutral (c) Quasi−neutral (d) Quasi−neutral

(e) Quasi−neutral

0.0

0.5

1.0

(f) Positive (g) Positive (h) Positive

(i) Quasi−neutral

0.0

0.5

1.0

(j) Positive (k) Negative (l) Negative

(m) Quasi−neutral

0 5000 10000

0.0

0.5

1.0

(n) Positive

0 5000 10000

(o) Positive

0 5000 10000

(p) Complex

0 5000 10000

Time

Reg

iona

l fre

quen

cyNeutral Fixed Independent Tradeoff

Figure 3: Regional frequency time series of a representative example for each outcome type.

In each panel, each line represents one of the ten replicate simulation run. Regional frequency

ranges for which priority effects are stronger than in the corresponding neutral scenario, i.e.

where the initially dominant species after a recolonization is more likely to remain dominant

than under neutrality, are highlighted in gray (see fig. A4 for more detailed information). Legend

continues on next page.

24


https://doi.org/10.1101/040659

Fig. 3 legend continued: Note that in (k) and (l), we do not have information on the strength of642

priority effects at extreme frequencies because they were not represented in the simulation runs,

but for all frequency ranges represented, priority effects were stronger than under neutrality. First644

row parameter values: k = 32, n = 294, hmax = 13.212, d = 0.05 · 2hmax, m = 0.161, ε = 0.017, f =

4, propagule-pool recolonization. Second row parameter values: k = 60, n = 355, hmax =646

14.55, d = 0.052 · 2hmax, m = 0.897, ε = 0.023, f = 5, propagule-pool recolonization. Third row

parameter values: k = 6, n = 461, hmax = 11.287, d = 0.023 · 2hmax, m = 0.101, ε = 0.015, f = 6,648

migrant-pool recolonization. Fourth row parameter values: k = 13, n = 1379, hmax = 12.393, d =

0.054 · 2hmax, m = 0.728, ε = 0.012, f = 4, propagule-pool recolonization. See fig. A2 for the650

corresponding frequency-dependence plots, which are the basis for classification.

25


https://doi.org/10.1101/040659

(a) Quasi−neutral

time

G

0.50

0.75

1.00

13.26.6

0

(b) Quasi−neutral

regional frequencyav

erag

e se

nsiti

vity

to g

ener

al c

ompe

titio

nG, i

n s

0.50

0.75

1.00

13.26.6

0

(c) Positive

time

G

0.50

0.75

1.00

14.67.3

0

(d) Positive

regional frequency

aver

age

sens

itivi

ty to

gen

eral

com

petit

ionG

, in

spec

ies

1

0.50

0.75

1.00

14.67.3

0

(e) Negative

time

G

0.50

0.75

1.00

11.35.6

0

(f) Negative

regional frequency

aver

age

sens

itivi

ty to

gen

eral

com

petit

ionG

, in

spec

ies

1

0.50

0.75

1.00

11.35.6

0

(g) Complex

time

G

0.50

0.75

1.00

11.95.9

0

0 5000 10000

(h) Complex

regional frequency

aver

age

sens

itivi

ty to

gen

eral

com

petit

ionG

, in

spec

ies

1

0.50

0.75

1.00

11.95.9

0

0.0 0.5 1.0

Time Regional frequency

Ave

rage

sen

sitiv

ity to

gen

eral

com

petit

ion,

G, i

n sp

ecie

s 1

Average sensitivity to species 2, H

2 , in species 1

0.3 0.4 0.5 0.6 0.7

0.88

0.90

0.92

0.94

1.35

1.13

0.9

0.68

Figure 4: Time series of the average values of the evolving traits in the focal species (species 1,

whose regional frequency is shown in fig. 3) for the example scenarios from fig. 3 under trade-

off evolution. The inset in (f) zooms in on the response of G to regional frequency. The same

colors are used for the corresponding replicates as in fig. 3. In (g), the upper, less variable lines

correspond to replicates where species 1 is the regionally rare species, whereas the lower, more

variable lines correspond to replicates where it is the regionally common species.


https://doi.org/10.1101/040659

(a)

(b)

(c)

(d)

0.2

0.4

0.6

0.8

1.0

Sta

nd

ard

devia

tion

of

muta

tion s

tep

s d

0.01

0.03

0.05

Avera

ge t

ime b

etw

een

dis

turb

ance

even

ts τ

2040

6080

100

Maxim

um

sensi

tivit

y t

o o

ther

speci

es h

max

4 6 8 10 12 14 16

05

1015

20

FrequencydependenceNegativeComplexQuasi−neutralPositive

Local community size k

Mig

rati

on r

ate

m

Figure 5: Effects of varying two parameters simultaneously while holding all other parameters

constant at the values from fig. 3 l. (a) Migration probability m and local community size k are

varied, (b) mutational standard deviation d and k are varied, (c) average time between distur-

bances τ and k are varied, (d) maximum sensitivity to heterospecific interference hmax and k are

varied.

27


https://doi.org/10.1101/040659

0 5000 10000 15000 20000

02

46

810

Time

Ave

rage

reg

iona

l spe

cies

ric

hnes

s

NeutralFixedIndependentTrade−off

Figure 6: Time series of regional species richness for different initial numbers of species. Each

point is the average over 10 replicate simulations. Parameters are the same as in the third row in

fig. 3.

28


https://doi.org/10.1101/040659

Online Appendix A: Supplementary methods and results652

Table A1: Model variables (capital letters) and parameters (small letters) and their default values

or sampling distributions.

Parameter

or variable

Explanation Default value/distributiona

Metacommunity level

s Number of species 2

n Number of patches lnN (ln(500), 0.5)

m Migration probability per birth event Beta(1,1)

Community level

Ni Number of individuals of species i in a local community

k Size of local communities blnN (ln(20), 0.5)eτ Average time between disturbance events lnN (ln(50), 0.5)

f Founder population size min(blnN (ln(5), 0.5)e, k)

Individual level

G General sensitivity to competition, i.e. from conspecifics

and heterospecifics

Gj Contribution to G due to resistance to species j

Hj Sensitivity to species j

gmin Minimum sensitivity to general competition 0.5

∆g Maximum per-species contribution to sensitivity to gen-

eral competition

0.5

hmax Maximum sensitivity to another species lnN (ln(10), 0.5)

g0 Initial value of the Gj 0.4b

d Standard deviation of mutation effect size (on the scale

of Gj)

lnN (ln(0.032), 0.5)

y Trade-off shape parameter 1

a lnN stands for the log-normal distribution. be denotes rounding to the nearest integer. b 0.25 in the

fixed-parameter scenario.

29


https://doi.org/10.1101/040659

Online Appendix A.1 Classification of regional frequency dependence

To assess regional frequency dependence, we divided the frequency interval between 0 and 0.5654

into 10 equally sized bins (from 0 to 0.05, from 0.05 to 0.10 etc.). After discarding the first few

sampling points (before time 50), we assigned each of the remaining ones to a frequency bin656

according to the regional frequency of the regionally less common species. For each bin, we then

computed the average change in regional frequency until the next sampling point, i.e. ten time658

units later. We did this for all replicates separately and then computed an interval of mean ±c

standard errors for each bin, where c is a positive constant.660

Based on these intervals, we assigned a simulated scenario to one of five regional frequency

dependence classes (fig. A1). If all intervals overlapped with zero, we assigned the scenario to662

the “quasi-neutral” class. After confirming independently that our neutral simulations indeed

exhibit neutral regional frequency dependence, we chose c such that 95% of neutral simulations664

were classified as quasi-neutral. We then used the resulting value c = 3.83 throughout the study.

All intervals overlapping 0?

Quasi-neutrality Most marginal interval that is not overlapping 0

Any negative intervals? Any positive intervals?

Complex frequencydependence

Negative frequencydependence

Inverse complex frequency dependence

Positive frequencydependence

yes no

>0 <0

yes no yes no

Regional frequency p

E[Δp

]

0 0.5

0

0 0.5

0

0 0.5

0

0 0.5

0

E[Δp

]

0 0.5

0

p

Figure A1: Classification algorithm based on the intervals of average regional frequency change,

E[∆p],± 3.83 standard errors for various regional species frequencies. Since our model is sym-

metric, we only need to consider the regional frequency of the species that is currently less

frequent in the metacommunity, i.e. we consider only regional frequencies between 0 and 0.5.

For scenarios that were not classified as quasi-neutral, we considered the most marginal fre-666

quency bin whose interval was not overlapping with zero (fig. A1). A positive average change

in this frequency bin indicates that a rare species can increase in frequency and that the regional668

species pool may be stable. Depending on whether or not there were any intervals with nega-

30


https://doi.org/10.1101/040659

tive average change, we assigned such scenarios either to the “complex frequency dependence”670

class or to the “negative frequency dependence” class. Under negative frequency dependence,

coexistence is expected to be symmetric, and under complex frequency dependence asymmet-672

ric with one species at high and one at low regional frequency. Analogously, scenarios whose

most marginal interval not overlapping zero was negative were assigned to either the “inverse674

complex frequency dependence” class or the “positive frequency dependence” class (fig. A1). In

both cases, a regionally rare species would be expected to go extinct, but under inverse complex676

frequency dependence two species that are initially equally common might coexist for a long

time. However, no parameter combination in our study exhibited inverse complex frequency678

dependence.

31


https://doi.org/10.1101/040659

Online Appendix A.2 Additional results for two species680

●● ● ● ● ●

● ● ● ●

(a) Quasi−neutral

midpoints

mea

nofm

eans

−0.

025

0.00

00.

025

● ● ● ● ●●

● ● ●●

(b) Quasi−neutral

midpoints

mea

nofm

eans

● ● ● ● ● ●● ● ● ●

(c) Quasi−neutral

midpoints

mea

nofm

eans

● ● ● ● ● ● ● ● ● ●

(d) Quasi−neutral

midpoints

mea

nofm

eans

● ●● ● ● ● ● ● ● ●

(e) Quasi−neutral

midpoints

mea

nofm

eans

−0.

025

0.00

00.

025

● ● ●●

● ●● ●

● ●

(f) Positive

midpoints

mea

nofm

eans

● ●● ● ● ● ●

●

●●

(g) Positive

midpoints

mea

nofm

eans

● ● ●● ● ● ● ● ●

●

(h) Positive

midpoints

mea

nofm

eans

●● ● ●

●●

●

● ● ●

(i) Quasi−neutral

midpoints

mea

nofm

eans

−0.

025

0.00

00.

025

● ● ●● ●

●● ●

● ●

(j) Positive

midpoints

mea

nofm

eans

●

●

●

●● ● ● ●

(k) Negative

midpoints

mea

nofm

eans

●

●●

●● ●

(l) Negative

midpoints

mea

nofm

eans

● ●●

● ●● ●

●

●

●

(m) Quasi−neutral

midpoints

mea

nofm

eans

0.00 0.25 0.50

−0.

025

0.00

00.

025

●

●●

● ●●

●

●

●●

(n) Positive

midpoints

mea

nofm

eans

0.00 0.25 0.50

●●

●● ●

● ●● ●

●

(o) Positive

midpoints

mea

nofm

eans

0.00 0.25 0.50

● ● ● ● ●● ● ● ●

●

(p) Complex

midpoints

mea

nofm

eans

0.00 0.25 0.50

Regional frequency

Mea

n ∆

regi

onal

freq

uenc

y

Neutral Fixed Independent Tradeoff

Figure A2: Frequency-dependence plots, i.e. intervals of mean regional frequency change ±3.83

standard errors for the various frequency bins between 0 and 0.5, corresponding to the the time

series in fig. 3.

32


https://doi.org/10.1101/040659

(a)

G

0.50

0.75

1.00

●● ●

●

●●

●●

●

● ●●

●●

●

● ●

●

●

●

●●

●●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●●

(b)

●

●

●

●

●

●

●

●●

●

●

●●●

●

●●

●

●

●

●●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●●

●

●●

● ●●●●

●●

●

●

●

●

●

●●

●

●

● ●●

●

●●

●

●

●

●●

●●●● ●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●

●●

●

●

● ●

●●●

●

●

●●

●

●

●●

●

●

●

●●

●●

●

●

●

●● ●

●●

●

●

●

● ●●

●●

●●●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

● ●

●

●

●

●

●●●

●

●●

●

●

●

●● ●

●

●●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

● ●●

●

●●

●

●●

● ●

●

●●●●

●

●●

●●●

●●

●●

●

●●

●

●●

●●● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●●

●

●●

●

●

●

●

●

●

●●

●●

●●●

●

●

●

●

●

●

●●●

●

●

●●●

●

● ●●

●● ●

●

●●

●

●

●●

●

●

●●

●

●●

●●

●

●

●

●

●●●●

●

●

●

●

●●

●●

●

●

●●

●

●●●●

●

●

●

●

●

●

●●

●●

●

●●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●●●

●●●

●

●●

●●

●

●●

● ●

●

●●

●

●●

●

●

●

●

●●

●● ●●● ● ● ●

●●●●

●●

●

●

●●

●●

●●

●●●

●

● ●

●

●

●

●

●

●

●

●●●

●

●

0.56

0.57

0.58

0.59

0.60

(c)

Time

H2

0 5000 10000

02

46

810

●

●

●

● ●

●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

● ●

● ●

●●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

0.2 0.3 0.4 0.5 0.6 0.7

0.9

1.1

1.3

1.5

(d)

Regional frequency

●

●

●●●

●●

●

●●

● ●

●

●●●

●

●●●

●●●

● ●

●

●

●

●

●●

●

●

●●●●

●●

●

●

●

● ● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●●●

●

●

●●

●●●

●●●

●

●●●

●

●

●●●●●

●

●

●●●

●●

● ●

●

●

●

● ●●

●

●

●

● ●●

●

●

●

●

●

●

● ●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●

●

●●●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●

●

●

●●

●●

●

●●●

●

●●●

●●

●●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●●● ●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●●●

●

●●

●

●

●●

●

●

●●

●●●● ●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●●●

●

●

●

●●

●●

●●

●●

●

●●

●●●

●

● ●●●

●●

●

●

●

●●

●●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●

●●●

●●

●

●●

●● ●

●●

●

● ●

●●

●

●

●●● ●●

● ●● ●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●●●

●

●

●

●

●

●

●

● ●

●●● ●

●●

●

●●

●

●

●

Figure A3: Example illustrating evolutionary dynamics under independent evolution. (a) Aver-

age sensitivity to general competition G for a focal species (species 1). (c) Corresponding average

sensitivity to heterospecific interference H2. (b) and (d) show how G and H, respectively, respond

to regional frequency. Parameters as in fig. 3 l (negative frequency dependence example).

33


https://doi.org/10.1101/040659

● ●● ● ●

●●

●●

●●

●

●●

●●

●● ● ●

●

(a) Quasi−neutral0.

500.

751.

00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

●

●

●

●


● ● ● ● ● ●●

●●

●●

●●

●

●

●

●

●●

● ●

(b) Positive

0.50

0.75

1.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●

●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

●●

●●

●●

●●

●●

●

●●

●

●● ●

● ● ●

●

(c) Negative

0.50

0.75

1.00 ●

●●

●●

●●

● ● ●●

● ● ●● ● ● ● ● ● ●

●● ● ● ●

● ●● ● ● ● ● ● ● ●

●

● ● ●● ● ● ● ● ● ● ●

●

● ● ● ● ● ●●

●●

●

●

●

●

●

●

●

●

● ● ●

(d) Complex

0.50

0.75

1.00

0.00 0.25 0.50

● ● ● ● ● ● ● ● ●● ● ●

●● ● ●

● ●●

●

● ● ● ● ● ● ● ● ●● ●

●●

● ●● ● ●

●

●

● ● ● ● ● ● ● ● ● ●●

● ●● ●

● ●● ●

●

Regional frequency

Str

engt

h of

prio

rity

effe

cts

Figure A4: Strength of priority effects for the representative examples from fig. 3, i.e. the prob-

ability that the locally dominant species directly after a disturbance event is also the dominant

species before the next disturbance event. The x axis represents the regional frequency of the re-

gionally rare species at the time of the disturbance event. The symbols for the trade-off scenario,

the fixed-traits scenario, and the independent-evolution scenario often overlap. In (c), coexistence

is so stable that regional frequencies never reach very marginal values and therefore we do not

have data on the strength of priority effects in this frequency range.


https://doi.org/10.1101/040659

Online Appendix A.3 Additional parameter sensitivity analyses for two species

Among the two outcome types that do not allow for stable coexistence, quasi-neutrality was682

more likely under propagule-pool recolonization and small migration rates, where the early in-

habitants of a patch mostly consist of just one species (fig. A5 e,h). Positive frequency dependence684

was more likely under migrant-pool recolonization and large migration rates, where the early

inhabitants of a patch are a more representative sample from the whole metacommunity (fig. A5686

e,h). This makes intuitive sense because in the first case species recolonize patches in proportion

to their regional frequencies, whereas in the second case the regionally common species wins688

disproportionately more often.

35


https://doi.org/10.1101/040659

●

●●●

●

●

●

●

●

●

1030

5070

(a)k ●

●

500

1000

1500

(b)

n●●●

●●

●

●●

●

●

1020

3040

(c)

hm

ax

●

●

●

0.02

0.06

0.10

(d)

d

●

●

●

0.0

0.4

0.8

(e)

m

●

●

●

●

●

●●

●

●

●

●

●

5010

015

020

0

(f)

τ

Com

plex

Neg

ativ

e

Pos

itive

Q.−

neut

ral

2

4

6

8

10

12

(g)

f

●

●

●

●

(h)

1:4

Fra

ctio

n pr

opag

ule−

pool

Com

plex

Neg

ativ

e

Pos

itive

Q.−

neut

ral

0.0

0.4

0.8

Figure A5: Distribution of parameter values for trade-off simulations classified as having com-

plex, negative, or positive frequency dependence, or quasi-neutrality. Since it is a binary param-

eter whether recolonization happens according to the propagule-pool or migrant-pool model, (h)

shows the fraction of simulations in each class that follow the propagule-pool model.

36


https://doi.org/10.1101/040659

0 5 10 15 20

0.0

0.1

0.2

0.3

0.4

0.5

(a)

Hj

Gj

y

0.215

4 6 8 10 12 14 16

0.1

0.5

2.0

5.0

(b)

k

y

NegativeComplexQuasi−neutralPositive

Figure A6: Role of the trade-off shape parameter y (see 2). (a) The trade-off is concave for y < 1,

linear for y = 1, and convex for y > 1. (b) Classification of parameter combinations with various

values of y and local community size k. All other parameters are held constant at the values from

fig. 3 l. Note the log-scale on the y axis.

37


https://doi.org/10.1101/040659

0.6 0.75 0.9

Total

Negative

Complex

Quasi−neutral

Positive

Fre

quen

cy−

depe

nden

ce c

lass

150 150 150

0 0 1

0 0 0

15 18 21

135 132 128 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

G

Proportion of coexistence

0

0.2

0.4

0.6

0.8

1

Figure A7: Coexistence and frequency dependence in the fixed-trait scenario with individual

traits fixed to three different points along the trade-off. The middle column corresponds to the

results in the main text. The parameter combination classified as having negative frequency

dependence is likely a misclassification.

38


https://doi.org/10.1101/040659

Online Appendix A.4 Additional results for more than two species690

Here we consider metacommunities of more than two species under a parameter combination

that exhibited complex frequency dependence with two species (last row in fig. 3). As in the692

negative frequency dependence example in fig. 6, metacommunities starting with three species

generally maintained all three species until the end. Metacommunities starting with four or five694

species maintained four species (fig. A8). However, metacommunities starting with ten species

often had only one or two species left at the end. To better understand this result, we performed696

two additional types of control simulation. In the “10-species” control, we started the metacom-

munity with only two species, but individuals had sensitivity traits Hj for 9 interacting species, as698

in the 10-species case. The two species in this control generally did not coexist until the end of the

simulation (fig. A8). In the “plasmid” control, we made the loci corresponding to extinct species700

no longer contribute to the overall competitive pressure. Qualitatively, this scenario represents a

situation where the relevant loci are on plasmids that can be rapidly discarded once they are no702

longer advantageous. With this modification, metacommunities starting at ten species stabilized

at four species (fig. A8). These additional results indicate that it is mutational noise at sensitiv-704

ity traits affecting interactions with extinct species that destabilized coexistence in the original

model.706

39


https://doi.org/10.1101/040659

0 5000 10000 15000 20000

02

46

810

Time

Ave

rage

reg

iona

l spe

cies

ric

hnes

s

NeutralFixedIndependentTrade−off10 lociPlasmid

Figure A8: Time series of regional species diversity (gamma diversity) for different initial num-

bers of species. Each point is the average over 10 replicate simulations. Parameters are the same

as in the last row in fig. 3.

40


https://doi.org/10.1101/040659

●●

●

●

●

●

●●

●

●●

●

●●

●●

●●

●

●●●●

●●●●

(a)

0.50

0.75

1.00

●

●●

●

●●

●●●

●●

●●●

●●●

●●●●

●●●●●●

●●

●●

●

●●

●●●●

●

●●●

●●●

●●●

●●●

●●

●●

●●

●

●●

●●

●

●

●


●●

●

●

●●

●

●●

●●

●●

●●

●●

●

●

●●●

●●●

●●

●●●

(b)

●●

●●

●●●

●●

●●

●●

●●●●

●●●

●●●●●

●●●●●

●●●

●

●

●

●●●

●●

●●

●●

●

●●●

●●

●●●●

●●●●●

●●

●●

●●

●●

●

●

●

●●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●●●

●●

●●

●

●●●

●●●

(c)

●

●

●●

●

●●●

●●

●●

●●

●●

●●●

●●●

●●●●

●●

●●●

●

●●

●●

●●●

●●

●●

●●●

●●

●●

●●

●●●

●●●●

●●●●

●●

●●

●●

●●

●

●●

●

●●●●

●

●●

●

●

●●

●

●

●

●●

●●

●

●

●●

●

●

●

●●●●

●

●●●

●●●●

●●●●

(d)

●

●

●

●

●

●

●●●●●●

●●●●

●

●●●

●●●

●●●●●

●●●●

●●●●

●●

●

●

●

●

●●●

●●●●●●●●●●●●●●●●●●●●

●●●●●●●

●●

●

●●

●●●●●●

●●●●●●

●●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●●●●●●●●●●

(e)

0.50

0.75

1.00

0.4 0.6 0.8 1.0

●

●●

●●

●●

●●●●

●●●●●●●●●●●●●●●●

●

●

●

●

●●

●●

●●

●●

●●●●●●●●●●●●●●●

●

●

●

●

●

●●

●●

●●

●●●●●●●●●●●●

●

●

●

●

●●

●

●●

●●

●

●

●

●

●

●●

●●●●●●●●●●●●

(f)

0.4 0.6 0.8 1.0

●

●

●●

●●

●●

●●

●●

●●●●●●

●●●●●●●●●●●●

●

●●

●

●

●

●●

●●

●●

●●●

●●●●●●●●●●●●●●●

●

●●

●

●

●●

●●

●●

●●

●●●●●●●●●●●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●●

●●●●●●●●●●●●●

(g)

0.2 0.4 0.6 0.8 1.0

●

●

●●

●●

●●

●●●

●●●●●

●●●●●●●●●●●●●●●●

●

●

●●

●

●●

●●

●●●

●●

●●●

●●●●●●●●●●●●●●●

●

●●

●

●●

●●

●●

●●

●●

●●●●●●●●●●●●●●●

●

●

●

●

●

●●

●

●●

●●

●

●

●

●

●

●

●●

●

●●●●●●●●●●●●●

(h)

0.2 0.4 0.6 0.8 1.0

●●

●●

●●●●●●

●●●●●●

●●●●

●●●●●●●●●●●●●●●●

●●●●●

●●

●

●●●●●●

●●●●●

●●●●●●●●●●●●●●●●●

●●●●●●●

●●●

●●●●●●●●●

●●●●●●●●●●●●●●●●●

Simpson's Index

Str

engt

h of

prio

rity

effe

cts

Figure A9: Strength of priority effects in the multi-species simulations. The first and second row

correspond to figs. 6 and A8, respectively. There are 3 species in the first column, four species in

the second column, five species in the third column, and ten species in the fourth column.

41


https://doi.org/10.1101/040659

Online Appendix A.5 Additional analysis on regional coexistence

In the following two sections, we will show that (1) in metacommunities with small local com-708

munities and a low migration rate, the regionally rare species tends to be “subjectively” common

locally, and (2) the more common a rare species is locally, the smaller is the difference in trait710

values between regionally common and rare species for the average member of the rare species

to be at an advantage in its local environment. The reason is that individuals do not suffer as712

much from heterospecific interference if they are locally common. From these two points, we can

conclude that with small community sizes and with small migration rates, a smaller difference714

in trait values is sufficient to tip the balance towards the regionally rare species. Thus negative

regional frequency dependence is more likely under smaller local community sizes and lower716

migration rates.

1. “Subjective” local frequency718

Consider a metacommunity where all patches have the same total community size k, as we

assume in our simulations. The mean local frequency of a species averaged over all patches720

is equal to its regional frequency. However, the mean local frequency might not adequately

portray the community composition experienced by the average individual of the focal species.722

For example, consider a metacommunity with a local community size of 10 and a focal species

at a regional frequency of 0.1. Let us also imagine that most patches are dominated by a single724

species as a result of local priority effects and that only a small fraction of patches contains both

species. In this scenario, the negative effects of interference by the common species will only726

be felt by a small fraction of individuals of the rare species. Had we assumed that most local

frequencies are close to the regional frequency, we would have erroneously concluded that most728

members of the rare species are in patches where the other species is locally common and that

they would therefore suffer strongly from the negative effects of interference by the other species.730

To formalize this intuition, we define the “subjective” local population size of a focal species

as the local population size experienced by the average individual of this species

E[N∗] := ∑ki=1 i2π(i)

∑ki=1 iπ(i)

, (A1)

where π(i) is the proportion of patches containing i individuals of the focal species. In other

words, when computing the average, patches are weighed by the local population size of the732

focal species. This way of averaging is formally known as size-biasing.

The mean subjective local population size can be easily computed from the mean and variance

42


https://doi.org/10.1101/040659

across patches of local population sizes N:

E[N∗] =E[N2]

E[N]=

E[N]2 + Var[N]

E[N]= E[N] +

Var[N]

E[N]. (A2)

As (A2) shows, the difference between size-biased expectation and un-biased expectation

increases with the variance in local community sizes. Intuitively, the larger variance in local

community sizes, the more heavily are patches dominated by a single species, and the larger is

the subjective local population size of the own species. To consider a specific example, let us

assume that local population sizes are binomially distributed with parameters p and k. We then

obtain

E[N∗] = pk + 1− p (A3)

and the subjective local frequency is

p∗ = p +1− p

k. (A4)

The smaller the local community size k is, the larger is the subjective local frequency for a given734

regional frequency (fig. A10 a). The increase in subjective local frequency compared to the re-

gional frequency in small patches can be substantial, especially for small regional frequencies.736

More generally

p∗ =E[N∗]

k= p +

Var[N]

k2 p. (A5)

Therefore, the average subjective frequency is a decreasing function of local community size

not only for the binomial distribution but whenever Var[N] increases with k not faster than k2.738

This appears to be the case in our simulated metacommunities. For small local communities,

the size-biased local frequency was substantially larger than the respective regional frequency740

(fig. A10 b). As a further intuitive illustration why subjective frequency should increase with

k, note that one is the minimum local abundance in which we can find members of the rare742

species. Thus, the smaller k is, the larger is the minimum possible local frequency of the rare

species. For example, if k = 4, the minimum possible local frequency is 0.25. In our simulations,744

the average subjective frequency decreased with migration rate (fig. A10 b), which makes sense

because frequent migration reduces variance in local species frequencies.746

2. Critical evolutionary response

Consider a rare species with sensitivity to general competition Gr and a common species with

sensitivity to general competition Gc, and let ∆G = Gr − Gc. The common species should evolve

to be a better intraspecific competitor or at least be under relaxed selection, so we expect ∆G > 0.

43


https://doi.org/10.1101/040659

20 40 60 80 100

0.0

0.2

0.4

0.6

(a)

Local community size

Ave

rage

sub

ject

ive

loca

l fre

quen

cy

●●

●● ● ● ●

● ● ● ● ●

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

(b)

Local community size

Ave

rage

sub

ject

ive

loca

l fre

quen

cy

●

●

●

●

●●

●●

● ● ●● ● ● ● ● ● ●

●●

●

●●

●● ●

● ● ● ● ● ● ● ● ● ● ●

●●

● ● ●● ● ●

● ● ● ● ● ●

●

●

●●

●● ●

● ● ● ● ● ● ● ● ● ● ●

●

●●

●●

●● ● ● ● ● ● ● ● ● ● ● ● ●

●

●

m

0.10.3

Figure A10: Average subjective frequency as a function of local community size k for different

regional frequencies. As the local community size increases, the average subjective frequency

approaches the respective regional frequency (indicated by horizontal lines). In (a) a binomial

distribution of local population sizes is assumed, (b) represents distributions of local community

sizes from our simulations, specifically those underlying fig. 5 a.

44


https://doi.org/10.1101/040659

Assuming a linear trade-off, we will now compute the critical value for ∆G such that the rare

species has an advantage locally when it has i individuals in a community of size k, with i < k/2.

Since the probability to reproduce is assumed to be the same for all species, a species has an

advantage locally when its members have a lower death rate Ci = G · k + H(G)(k− Ni) with

H(G) =

(1− G− gmin

∆g

)· hmax. (A6)

In our study, we used gmin = 0.5 and ∆g = 0.5 so that

H(G) = 2(1− G) · hmax. (A7)

With this, the death rate for rare-species individuals is

Cr = Grk + 2(1− Gr)hmax(k− i) (A8)

and the death rate for members of the common species is

Cc = Gck + 2(1− Gc)hmaxi = (Gr − ∆G)k + 2(1− Gr + ∆G)hmaxi. (A9)

Members of the rare species have an advantage, i.e. a lower death rate, if

Gr(k− 2hmax(k− i)) + 2hmax(k− i) < (Gr − ∆G)(k− 2hmaxi) + 2hmaxi (A10)

⇔ Gr(−2hmax(k− 2i)) + 2hmax(k− 2i) < ∆G(2hmaxi− k) (A11)

⇔ 2hmax(k− 2i)(1− Gr) < ∆G(2hmaxi− k). (A12)

We need to consider two cases:748

a) k > 2hmaxi. Since i < k/2, it follows from (A12) that

∆G <2hmax(k− 2i)(1− Gr)

2hmaxi− k< 0. (A13)

b) k < 2hmaxi750

∆G >2hmax(k− 2i)(1− Gr)

2hmaxi− k> 0. (A14)

Since the common species will generally have a lower G and ∆G > 0, (A13) cannot be fulfilled.

If hmax < 1 then k > 2hmaxi for all i < k/2, and it would be impossible for a locally rare752

species to have a local advantage, no matter how sensitive the common species is to heterospecific

interference. This should prevent regional coexistence. However, in our simulations we did not754

consider such low values of hmax.

45


https://doi.org/10.1101/040659

In case b), we get a critical trait difference

Rcrit :=2hmax(k− 2i)(1− Gr)

2hmaxi− k(A15)

such that a globally and locally rare species has a local advantage whenever the difference in G756

values is at least Rcrit. The smaller Rcrit is, the easier it is for evolution along the trade-off to

stabilize regional coexistence.758

The critical trait difference is a function of local population size i and

∂Rcrit

∂i= 2hmax(1− Gr)

−2(2hmaxi− k)− (k− 2i)2hmax

(2hmaxi− k)2 =4(1− Gr)khmax(1− hmax)

(2hmaxi− k)2 . (A16)

If hmax > 1, which is the case if the condition of case b) is fulfilled and k > 1, then the critical

evolutionary response necessary to give an advantage to the rare species decreases with its local760

population size i. As explained in the main text and illustrated in fig. A11, this can give an

average advantage to the rare species while at the same time priority effects are maintained, but762

slightly asymmetric. However, not all values of k have the same potential for such slight asymme-

try. For odd k, for example, there are no patches with a completely balanced species composition764

and to start having a regional advantage the regionally rare species needs to dominate even in

patches where the regionally common species is slightly more abundant, which presumably re-766

quires a larger evolutionary response. Such “discreteness effects” may underlie the idiosyncratic

relationship between k and outcome type in fig. 5 a.768

∂Rcrit

∂hmax= 2(k− 2i)(1− Gr)

2hmaxi− k− 2hmaxi(2hmaxi− k)2 =

−2k(k− 2i)(1− Gr)

(2hmaxi− k)2 < 0. (A17)

Therefore, with increasing hmax the evolutionary response necessary to provide an advantage to

the regionally rare species decreases. This is consistent with the finding in fig. 5 g: regional770

negative frequency dependence occurred only for sufficiently large hmax. Under a high hmax, the

common species is punished heavily for neglecting to invest in heterospecific tolerance.772

46


https://doi.org/10.1101/040659

1 2 3 4 5

(a) Small trait difference

0

1 2 3 4 5

(b) Large trait difference

Local abundance of regionally rare species

Rel

ativ

e lo

cal f

itnes

s of

reg

iona

lly r

are

spec

ies

Figure A11: Cartoon illustrating a situation with priority effects at extremely unbalanced local

species configurations and an advantage of the regionally rare species under more balanced

configurations. The local community size k is 6. (a) Small evolutionary response, ∆G. (b) Large

evolutionary response, ∆G.

47


https://doi.org/10.1101/040659

Documents

Eco-evolutionary buffering: rapid evolution facilitates ...€¦ · 6/3/2016 · 26 2012; Fukami and Nakajima 2013). Founded in large part on MacArthur and Wilson’s (1967) theory